How is instrumental variables with many exogenous predictors related to propensity scoring? In a large (many exogenous predictors) instrumental variables regression, with covariate matrix $X$, outcome $Y$ and binary treatment indicator $T$, a two-stage least-squares approach might be like this:
$ logit(P(T = 1)) = X\beta + error$ (logistic regression of receiving treatment)
$ P(T = 1)^{fit} = invlogit \{ X\beta^{fit} \}$
$Y = \gamma_{effect}P(T = 1)^{fit} + error$
where the instrument(s) are included in $X$ in the first stage regression, along with all other covariates, assuming that non-instrument covariates are exogenous. (Note that even if the instruments are retained in $X$ for the second stage, by the assumptions that the instrument only affects the outcome through the treatment, it should have no impact on parameter estimation in the second regression stage.)
My question is this: won't $P(T = 1)^{fit}$ be equivalent to a propensity score?
The second-stage regression will be different than weighting $X$ by the propensity score and including regular $T$, but I am trying to understand exactly how they will differ, and whether there is a connection between the IV case when you need to basically include the whole design matrix in the first stage regression because it contains so many exogenous variables, vs. when you estimate propensity scores and weight by them.
 A: Usually when someone says that they are using propensity scores, what they mean is that they are using propensity score matching.  The idea in propensity score matching is to take two groups of observations with the same propensity score but differing on treatment.  I.e. the idea is to compare treatment vs control while controlling for the propensity score.  
The idea of instrumental variables is kind of the opposite.  What you are trying to do with instrumental variables is to ignore actual treatment and just look at how outcomes vary based on predicted treatment given the instrumental variables.
A simple/crude way to use propensity scores to estimate treatment effects would be to run the following regression by OLS:
\begin{align}
Y = T^{\text{fit}} \gamma + T \delta + \epsilon
\end{align}
Then you would call $\hat{\delta}_{\text{OLS}}$ your estimate of the treatment effect.
A similarly simple/crude use of instrumental variables would be to run the following regression by OLS:
\begin{align}
Y = T^{\text{fit}} \alpha + \epsilon
\end{align}
Then you would call $\hat{\alpha}_{\text{OLS}}$ your estimate of the treatment effect.
These are basically opposite strategies for estimating the effect of the treatment.  The instrumental variables estimator looks only at variation in treatment induced by $X$: ie at $T^{\text{fit}}$.  The PS matching estimator looks at variation in treatment, but removing variation in treatment induced by $X$.
Imagine parceling out the variation in treatment into two components: that explained by $X$ and that not explained by $X$.  The IV approach considers the variation explained by $X$ to be "good" variation and variation not explained by $X$ as "bad" variation.  The PS matching approach considers the variation explained by $X$ to be "bad" variation and variation not explained by $X$ as "good" variation.
If you ask, under what statistical assumptions is each of the estimators good, then you will see this distinction repeated.  PS matching estimators are good estimators when treatment, after controlling for $X$, is exogenous.  IV estimators are good estimators when $X$ is exogenous (has no direct effect on outcomes).
Whether each procedure works to reveal true treatment effects depends on what is true about the causal and statistical relationships in your application.   
You might be interested in a paper I wrote with a colleague which is closely related to this point.
